Complex Numbers MAΘ 2013-14

What is a Complex Number? i is the square root of -1 Form: a + bi Conjugate of a + bi is a - bi In the complex plane, the x-axis has real numbers and the y-axis has purely complex numbers

Using Complex Numbers n solutions to any degree n polynomial What is ? What about ? And ? Solve these two equations What solutions do we know for Challenge (don’t try): what is

Common Techniques Each complex number has an imaginary and a real part -- you can usually get two equations out of this. Example: (a + bi)(c + di) is real. What does that mean about a, b, c, and d?

Imaginary Roots Graph of Only crosses x-axis twice -- only two real roots Others are imaginary Notice that the imaginary roots come in conjugate pairs.

Polar Form Can express as What is i in this notation? How do we go from rectangular, (a+bi), to polar, (r,Θ)?

Multiplying Them Multiply Magnitudes, Add Angles! So what is How do we use this to find given Find -- with two ways

Exponential Form Given z = a + bi, factor out r, from (r,Θ). z = r(cos Θ + i sin Θ) Euler’s Formula states that Therefore z = reiΘ

Roots of Unity What are the solutions of ? Let x = (r,Θ). Then x9 = (r9, 9Θ), and 1 = (1, 0 + 2πk). These two are equal if r9 = 1 or 9Θ = 2πk. r must be a positive integer by definition, so r = 1, and Θ = 2πk/9. Thus x = (1, 2πk/9), where k = 0, ±1, ±2, etc.